As Patman stated there are a few reasons for the condition you are experiencing. It would be nice to have a picture of the engine (clutch side) and I could help provide some possible solutions.
First adjust the clutch properly. This will vary depending on what type of clutch you have. There are 3 basic styles Semi-Auto, Crank mounted clutch, and primary/transmission mounted clutch. If you don't know how to adjust the clutch I can type out a detailed procedure. This will cure the first problem with the lurching when shifting from neutral into 1st gear.
The second problem can be exacerbated by not correcting the first problem.
There are two fairly easy and inexpensive "fixes" you can try, but these involve removing the clutch cover and a few components.
First the most inexpensive fix (costing less than $5 and your labor) is replacing the shift star detent arm spring. Its fairly common (it became the problem on 2 different motors I have had) that the "stock" spring looses its temper or is weak and does not fully allow the detent arm to fully engage the shift star, thus allowing it to turn and change gears at will. The spring is available from any Honda dealer or distributor of OEM Honda parts. Below are some pictures for reference and part number.
Spring P/N the Honda spring is stiffer
http://i33.photobucket.com/albums/d80/princevaliant/10-20-07_1255.jpg
Crank mounted clutch type motor
http://i33.photobucket.com/albums/d80/princevaliant/IMG_3404-1.jpg
Transmission/primary mounted clutch type motor
http://i33.photobucket.com/albums/d80/princevaliant/10-27-07_1657.jpg
(clutch removed for clarity of the shift star and detent arm)
http://i33.photobucket.com/albums/d80/princevaliant/GPXside.jpg
The next thing to do/try is modify your existing shift star by grinding down the valleys and making the peaks more steep, thus a more positive engagement of the deten arm. The other solution along the same lines is replace it with a corresponding OEM Honda part. This is where some trial and error will come into play as you will have to order a shift star from several models to find what style they used since Honda changed the design ever so slightly from model to model on their horizontal engines. Try models like: TRX90, CT/XL/SL70, CT90/110.
If all that fails to fix the problem then the problem lies within the (worn) shift drum or forks, they are replaceable with the corresponding Honda parts, but in most cases it is just easier to buy a (known quality, Lifan, Jialing, GPX, Zongshen) crate engine and drop that in, such as these.
http://tboltusa.com/store/index.php?cPath=114
or these.
http://www.sgr-usa.com/index.php?cPath=153&osCsid=f68b39d5add5f5b66b43fe78f698467d
As well as other sources. However the two sources I list are reputable and will give tech support and or limited warranties.